We derive an explicit count for the number of singular n × n Hankel (Toeplitz) matrices whose entries range over a finite field with q elements by observing the execution of the Berlekamp/ Massey algorithm on its elements. Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed. For example, the number of singular n × n Toeplitz matrices with ...

In this paper, we present four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture. We particularly emphasize Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left-and right-singular vectors) for sparse matrices arising from two pra...

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Let complex matrices $A$ and $B$ have the same sizes. Using the singular value decomposition, we characterize the $g$-inverse $B^{(1)}$ of $B$ such that the distance between a given $g$-inverse of $A$ and the set of all $g$-inverses of the matrix $B$ reaches minimum under the unitarily invariant norm. With this result, we derive additive and multiplicative perturbation bounds of the nearest per...

It is well known that any nonsingular M–matrix admits an LU factorization into M–matrices (with L and U lower and upper triangular respectively) and any singular M–matrix is permutation similar to an M–matrix which admits an LU factorization into M–matrices. Varga and Cai establish necessary and sufficient conditions for a singular M–matrix (without permutation) to allow an LU factorization wit...

In this paper, a method for through-wall human detection based on the singular values decomposition of the measurement matrices is presented. After demonstrating the sparsity of the matrices using CLEAN algorithm, an SVD algorithm based on Lanczos process is applied to compute their singular values. We also analyze the singular values of matrices constructed by difference square techniques for ...

In this paper we derive a new algorithm for constructing a uni-tary decomposition of a sequence of matrices in product or quotient form. The unitary decomposition requires only unitary left and right transformations on the individual matrices and amounts to computing the generalized singular value decomposition of the sequence. The proposed algorithm is related to the classical Golub-Kahan proc...

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P(s)(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P(s)(x) in terms of a combination of s hypergeometric functions of the type (s)F(s-1). The explicit formula derived here...

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all non commutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically fr...

For matrices with a well defined numerical rank in the sense that there is a large gap in the singular value spectrum we compare three rank revealing QR algorithms and four rank revealing LU algorithms with the singular value decomposition. The fastest algorithms are those that construct LU factorizations using rook pivoting. For matrices with a sufficiently large gap in the singular values all...